In Algebra 1 you used residuals to judge a line of fit. With the least-squares line they become a powerful check: a plot of the residuals shows whether a line was the right model at all.
Key idea
The residual of a point is $e = y - \hat{y}$ (actual minus predicted). A residual plot graphs the residuals against x. If a line is a good model, the residuals look like random scatter around 0 with about the same spread everywhere. A curve or a fan shape means the line is missing something.
Why it works
The least-squares line already removes the straight-line trend, so whatever is left over is what the line could not explain. If that leftover still has a pattern, such as positive at both ends and negative in the middle, the data bend and a straight line systematically misses. A random cloud means there is nothing systematic left for a line to fix.
Worked example: One residual
The line is $\hat{y} = 57 + 3x$. A student who studied 8 hours scored 85. Find the residual.
- Predicted: $\hat{y} = 57 + 3(8) = 81$.
- Residual: $85 - 81 = 4$.
- Positive: the point lies 4 points above the line.
Answer: $e = 4$
Worked example: Reading a residual plot
For $x = 1, \ldots, 10$ and $y = x^2$, the line has $r \approx 0.97$. Its residual plot is shown above. Is a line a good model?
- The residuals form a U: positive, then negative, then positive again.
- That pattern means the line is too high in the middle and too low at the ends: the data curve.
- So despite the high $r$, a curved model would fit better.
Answer: No. The U-shaped residual plot shows the form is curved.
Common mistake
Taking a high $r$ as proof that a linear model is right. $r$ measures how close the points are to some line; it does not say a line is the right shape. Check the residual plot: any curve or fan means "not linear" no matter how big $r$ is.